Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, and equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}(r)d\theta^{2},$$ where $dr^{2}$ is the standard metric on $\mathbb{R}_{+}$ and $d\theta^{2}$ is the standard metric on the sphere $\mathbb{S}^{n-1}$. Denote with $\mu$ the Riemannian measure on $\mathbb{H}^{n}$ and with $\sigma$ the Riemannian measure of co-dimension $1$ on hypersurfaces on $\mathbb{H}^{n}$. It is a known fact that $\mathbb{H}^{n}$ admits the Isoperimetric inequality $$\sigma(\partial \Omega)\geq f(\mu(\Omega)),$$ for all precompact open sets $\Omega\subset \mathbb{H}^{n}$ with smooth boundary, where $f$ is defined by $$f(v)=c_{1}\left\{\begin{array}{cl}v, &v\geq 1\\ v^{\frac{n-1}{n}}, &v\leq 1,\end{array}\right.$$ for some small constant $c_{1}>0$.

Let us fix some precompact open set $U\subset \mathbb{H}^{n}$ and consider $\mathbb{H}^{n}\setminus U$ as a manifold with boundary $\partial U$.

Now my **question**:

Does the same Isoperimetric inequality now hold for precompact open sets $\Omega\subset \mathbb{H}^{n}\setminus U$ with smooth boundary (and possibly a smaller constant $c_{2}>0$) and if yes, where can I find a reference for this statement?

**Specified question**:

Let $U$ be a "nice" precompact open set, for example $U=B_{1}(o)$ is an open ball of radius $1$ for some point $o\in \mathbb{H}^{n}$. Does the same Isoperimetric inequality now hold for precompact open sets $\Omega\subset \mathbb{H}^{n}\setminus B_{1}(o)$ with smooth boundary (and possibly a smaller constant $c_{2}>0$)?

Note that we assume that, when considering a precompact open set $\Omega\subset \mathbb{H}^{n}\setminus U$ with smooth boundary, we have $\partial U\cap \partial \Omega=\emptyset$.

If this is not known for the hyperbolic space $\mathbb{H}^{n}$, is a similar statement known for other Riemannian manifolds, for example $\mathbb{R}^{n}$?

Thanks in advance for your help!

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